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Note on the Chebyshev polynomials and applications to the Fibonacci numbers. (English) Zbl 0844.11012

Let \((T_n (x))_{n\geq 0}\) be the sequence of Chebyshev polynomials of the first kind defined by \(T_{n+2} (x)= 2x T_{n+1} (x)- T_n (x)\) with \(T_0 (x)=1\) and \(T_1 (x)=x\). For these polynomials it is shown that the product of any two distinct elements of the set \(\{T_n, T_{n+2r}, T_{n+4r}, 4T_{n+r} T_{n+2r} T_{n+3r}\}\), increased by \(({1\over 2} (T_h- T_k))^t\), where \(t\) (\(=1\) or 2) and \(k> h\geq 0\) are suitable integers, is a perfect square for any natural numbers \(n\) and \(r\). Applications and generalizations of the result are also shown.
Reviewer: Péter Kiss (Eger)

MSC:

11B39 Fibonacci and Lucas numbers and polynomials and generalizations
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